NATURE 



[June I, 1905 



very slowly and those for which the rate is appreciable ; 

 but as c-N varies rapidly with N when N is large, there 

 will be but few vibrations near the border, so that it seems 

 legitimate, for purposes of a general discussion, to divide 

 the vibrations into the two distinct classes, quick and 

 slow, relatively to the scale of time provided by molecular 

 collisions. 



When the material bodies are solid, the physical prin- 

 ciple is the same, the relatively slow motions of the atoms 

 affecting the " quick " vibrations of the ether only by 

 raising a sort of " equilibrium tide." 



The number of " slow " vibrations of the ether in any 

 finite enclosure is finite. These quickly receive the energy 

 allotted to them by the theorem of equipartition. Thus 

 they form the medium of transfer of radiant energy 

 between two bodies at different temperatures. After a 

 moderate time the slow vibrations have each, on the 

 average, energy equal to that of two degrees of trans- 

 lational freedom of one molecule ; the quick vibrations 

 have no appreciable energy, while the intermediate vibra- 

 tions possess some energy, but not their full share. It 

 is easily seen that the number of slow vibrations is 

 approximately proportional to the volume of the enclosure, 

 so that roughly the energy of ether must be measured 

 per unit volume in order to be independent of the size of 

 the enclosure. For air under normal conditions, I find as 

 the result of a brief calculation that this value is of the 

 order of 5x10-' times that of the matter. The law of 

 distribution of this energy will be 



until we arrive at values of A which are so small as to 

 be comparable with 



,. , , , velocity of light 



radms of molecule X — , — : —2 ^ 



velocity of molecule 



After these values of A are passed, the formula must be 

 modified by the introduction of a multiplying factor which 

 falls off very rapidly as K decreases, and which involves 

 the time during which the gas has been shut up. It is 

 easily found (c/. "The Dynamical Theory of Gases," 

 g 247) that at 0° C. the spectrum of radiant energy is 

 entirely in the infra-red ; at 28,000° C. it certainly extends 

 to the ultra-violet, and probably does so at lower 

 temperatures. 



Finally, Lord Rayleigh asks : — 



" Does the postulated slowness of transformation really 

 obtain? Red light falling upon the blackened face of a 

 thermopile is absorbed, and the instrument rapidly in- 

 dicates a rise of temperature. Vibrational energy is 

 readily converted into translational energy. Why, then, 

 does the thermopile itself not shine in the dark?"' 



Before trying to answer this, I wish to emphasise that 

 my position does not require the forces of interaction 

 between matter and ether to be small. Considering a gas 

 for simplicity, the transfer of energy per collision to a 

 vibration of frequency p is found to be proportional to the 

 square of the modulus of an integral of the form (cf. " The 

 Dynamical Theory of Gases," g 237) 



where /(() is a generalised force between matter and ether. 

 The integral may be very small either through the small- 

 ness of /(() or the largeness of p. I rely entirely on the 

 largeness of p, because calculation shows this to be 

 adequate. The thermopile experiment gives evidence as to 

 the magnitude of /{(). but this does not alter the fact that 

 the integral is small for large values of p. 



This being so, I am afraid I do not very clearly under- 

 stand why the thermopile should be expected to shine in 

 the dark. If the red light is a plane monochromatic wave, 

 its energy represents only two coordinates of the ether', 

 and has to be shared between the great number of co- 

 ordinates, six for each atom, which belong to the thermo- 

 pile. If the red light comes from a large mass of red- 

 hot matter inside the same enclosure as the thermopile, 

 then the thermopile will soon be raised to the tempera- 

 ture of this mass, and may shine in the dark. If the hot 

 mass consists of iron, say at 600° C, the atomic motions 

 m the iron must be sufficiently rapid to excite the red 

 NO. 1857, VOL. 72] 



vibrations in the ether. But if the face of the thermo- 

 pile is of lampblack, the atomic motions in lampblack at 

 600° C. may not be of sufficient rapidity (mainly, so far as 

 can be seen, on account of the lower elasticity of the 

 material) to excite red vibrations except as a kind of 

 " equilibrium tide," in which case the lampblack will not 

 emit red radiation. 



I cannot ask for further space in which to answer Lord 

 Rayleigh 's point as to the enclosure with a hole in it, but 

 I have discussed a similar question in a paper which I 

 hope will soon be published, in connection with Bartoli's 

 proof of Stefan's law. I hope that this paper, and a 

 second one which is at present in the hands of the printer, 

 will explain my position more clearly than I have been 

 able to in the short limits of a letter. 



May 20. J. H. Jeans. 



Fictitious Problems in Mathematics. 



1 HAVE to thank your reviewer for so readily supplying 

 (Nature, May 18, p. 56) the example to prove his conten- 

 tion — and which appears (to me) to disprove it. 



The man who set that example did so in order to test 

 (inter alia) whether the pupil knew that, for any friction 

 to arise, both the surfaces must be rough ; your reviewer 

 originally wrote : — " What the average college don forgets 

 is that roughness or smoothness are matters which con- 

 cern two surfaces not one body." The italics are vour 

 reviewer's; and this is the statement which I called (and 

 still call) in question. 



It is no part of my book to uphold the verbiage in 

 which the example is couched ; by chance, in my former 

 letter, I e.xplained in anticipation the terms used in it. I 

 do not see, however, why your reviewer applies the 

 favourite word inaccurate to these terms. Perfect smooth- 

 ness may not occur in nature ; still, in considering the 

 pendulum, I probably begin by assuming no friction on 

 the axis of suspension, and. If I try afterwards to apply 

 a correction for this friction, I probably make an assump- 

 tion which is inaccurate. Friction = pressure x a constant 

 is inaccurate, statically and dynamically. 



C. B. Clarke. 



.^s I take it, the mathematician's " perfectly rough 

 body " means a body which never by any chance slips on 

 any other body with which it is placed in contact, similarly 

 the " perfectly smooth body " is supposed never to offer 

 any tangential resistance to any other body which it 

 touches. The inconsistency of this nomenclature is evident 

 when we imagine the two bodies placed in contact with 

 each other, as in the case of the perfectly rough plank 

 resting on the smooth horizontal plane. The subsequent 

 course of events cannot at the same time be compatible 

 with the assumed perfect roughness of the one body and 

 the assumed perfect smoothness of the other. The co- 

 efficient of friction between two bodies depends essentially 

 on the nature of the parts of the surfaces of both bodies 

 which are in contact as well as on their lubrication, and 

 neither body can be said to have a coefficient of friction 

 apart from the other. It is equally incorrect to speak of 

 perfect smoothness or perfect roughness as attributes of a 

 single body. Moreover, this misleading language is quite 

 unnecessary; it is very easy to frame questions in a way 

 that is free from objection. For instance, " A man walks 



I 



without slipping along a plank which can slip without 

 friction on a horizontal table." Or again, "A sphere is 

 placed in perfectly rough contact with the slanting face 

 of a wedge whose base rests in perfectlv smooth contact 

 with a horizontal plane." G. H. Bryan. 



A New Slide Rule. 



In the article which appeared on p. 45 of Xatl-re, 

 May II, describing the Jackson-Davis double slide rulel 

 you notice one little fault in the rule sent for examination. 



We desire to exonerate the designer of the instrument, 

 Mr. C. S. Jackson, from responsibility for the very obvious 

 fault to which you allude, viz. that the scale on the feather 

 edge is divided into inches and sixteenths, and that the 

 continuation scale which is read below the ordinary slide 



